Derivative of y=log3(x)-sqr(xx^5)

1. Differentiate $- \left(x x^{5}\right)^{2} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}}$ term by term:

   1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$.

      So, the result is: $\frac{1}{x \log{\left(3 \right)}}$

   2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Let $u = x x^{5}$.

      2. Apply the power rule: $u^{2}$ goes to $2 u$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x x^{5}$:

         1. Apply the product rule:

            $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

            $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

            1. Apply the power rule: $x$ goes to $1$

            $g{\left(x \right)} = x^{5}$; to find $\frac{d}{d x} g{\left(x \right)}$:

            1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$

            The result is: $6 x^{5}$

         The result of the chain rule is:

         $$12 x^{11}$$

      So, the result is: $- 12 x^{11}$

   The result is: $- 12 x^{11} + \frac{1}{x \log{\left(3 \right)}}$

The answer is:

$$- 12 x^{11} + \frac{1}{x \log{\left(3 \right)}}$$